L(s) = 1 | + (−0.507 + 0.507i)2-s + (0.866 + 0.5i)3-s + 1.48i·4-s + (1.10 − 0.295i)5-s + (−0.693 + 0.185i)6-s + (0.718 + 2.54i)7-s + (−1.76 − 1.76i)8-s + (0.499 + 0.866i)9-s + (−0.409 + 0.709i)10-s + (1.72 − 0.462i)11-s + (−0.741 + 1.28i)12-s + (−3.60 − 0.189i)13-s + (−1.65 − 0.928i)14-s + (1.10 + 0.295i)15-s − 1.16·16-s + 3.69·17-s + ⋯ |
L(s) = 1 | + (−0.359 + 0.359i)2-s + (0.499 + 0.288i)3-s + 0.741i·4-s + (0.492 − 0.131i)5-s + (−0.283 + 0.0759i)6-s + (0.271 + 0.962i)7-s + (−0.625 − 0.625i)8-s + (0.166 + 0.288i)9-s + (−0.129 + 0.224i)10-s + (0.520 − 0.139i)11-s + (−0.214 + 0.370i)12-s + (−0.998 − 0.0525i)13-s + (−0.443 − 0.248i)14-s + (0.284 + 0.0761i)15-s − 0.292·16-s + 0.895·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0699 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0699 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.894575 + 0.959497i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.894575 + 0.959497i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.718 - 2.54i)T \) |
| 13 | \( 1 + (3.60 + 0.189i)T \) |
good | 2 | \( 1 + (0.507 - 0.507i)T - 2iT^{2} \) |
| 5 | \( 1 + (-1.10 + 0.295i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.72 + 0.462i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 + (-0.160 + 0.600i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 2.26iT - 23T^{2} \) |
| 29 | \( 1 + (0.0743 + 0.128i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.24 + 4.64i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-3.17 - 3.17i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.12 + 4.19i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.81 - 3.93i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0779 + 0.291i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.19 + 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.0 + 10.0i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.51 + 1.45i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.12 + 11.6i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (3.67 + 13.6i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-11.3 - 3.05i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.537 - 0.930i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.90 - 3.90i)T + 83iT^{2} \) |
| 89 | \( 1 + (8.17 - 8.17i)T - 89iT^{2} \) |
| 97 | \( 1 + (-4.98 + 1.33i)T + (84.0 - 48.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.18432850883324086397837801741, −11.36357572390073140247754107357, −9.641248690105021559680666806309, −9.422425352482348622050193009925, −8.259113669776991299081295430087, −7.58027451182806739100360068641, −6.24833139199074422445730510702, −5.05567612630377491142481956314, −3.56853858088592192350894638880, −2.30829772882960970147189735221,
1.21348801148057270915944813068, 2.56619837157546278476575119970, 4.28105424496889933686373753917, 5.63884739060791536275782932090, 6.81026552035240578965395980975, 7.78224581525276544378710140571, 9.014139508389165807144350933523, 9.923297670403308980558986877399, 10.39506249756449448773686528814, 11.56478953468947564264401151577